Question

The radius of a circle is 9 cm. Find the length of an arc of this circle which cuts off a chord of length, equal to length of radius.

Answer 1

Here, r = 9 cm

Let AB be the chord of the circle with centre at O such that l(chord AB) = 9 cm. Let OM be the perpendicular drawn from the centre O to the chord AB.

Then M is the midpoint of AB.

∴ `l("AM") = 1/2 × l ("AB") = 9/2 = 1/2 × l ("OA")`

Let m∠AOM = θ₁. 

Then in right-angled triangle OMA, 

sin θ₁ = `"AM"/"OA" = ((9/2))/9 = 1/2`

∴ sin θ₁ = sin 30°

∴ θ₁ = 30°

∴ m∠AOB = θ = 2·m∠AOM= 2θ₁ 

∴ θ = 60° = `(60 × π/180)^"c" = π^"c"/3`

∴ length of the arc = S = r·θ = 9 × `π/3` = 3π cm.